Joakim Arnlind, Martin Bordemann, Jens Hoppe, Choonkyu Lee
We relate free vector dynamics to the eigenvalue motion of a time-dependent real-symmetric NxN matrix, and give a geodesic interpretation to Ruijsenaars Schneider models.
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We discuss a general approach permitting the identification of a broad class of sets of Poisson-commuting Hamiltonians, which are integrable in the sense of Liouville. It is shown that all such Hamiltonians can be solved explicitly by a separation of variables {\em Ansatz}. The method leads in particular to a proof that the so-called "goldfish" Hamiltonian is maximally superintegrable, and leads to an elementary identification of a full set of integrals of motion. The Hamiltonians in involution with the "goldfish" Hamiltonian are also explicitly integrated. New integrable Hamiltonians are iden…
Ruijsenaars duality for B, C, D Toda chains
本文利用Hamiltonian约化方法构造了B、C、D型Toda链的Ruijsenaars对偶系统，证明其为rational Goldfish model的类似物，并给出了对偶Hamiltonian公式及高阶Hamiltonian的显式计算。
By formally comparing the geodesic equation with the Schrödinger equation on Riemannian manifold, we come up with the geometric Hamiltonian matrix on Riemannian manifold based on the geospin matrix, and we discuss its eigenvalue equation as well. Meanwhile, we get the geometric Hamiltonian function only related to the scalar curvature.
A supersymmetric extension of the Hunter-Saxton equation is constructed. We present its bi-Hamiltonian structure and show that it arises geometrically as a geodesic equation on the space of superdiffeomorphisms of the circle that leave a point fixed endowed with a right-invariant metric.
The reduced Hamiltonian system on T*SU(3)/SU(2)) is derived from a Riemannian geodesic motion on the SU(3) group manifold parameterised by the generalised Euler angles and endowed with a bi-invariant metric. Our calculations show that the metric defined by the derived reduced Hamiltonian flow on the orbit space SU(3)/SU(2)=S^5 is not isometric or even geodesically equivalent to the standard Riemannian metric on the five-sphere S^5 embedded into R^6.
The Relationship between the Neumann system and the Jacobi system in arbitrary dimensions is elucidated from the point of view of constrained Hamiltonian systems. Dirac brackets for canonical variables of both systems are derived from the constrained Hamiltonians. The geodesic equations corresponding to the Rosochatius system are studied as an application of our method. As a consequence a new class of nonlinear integrable equations is derived along with their conserved quantities.
We present a unifying variational calculus derivation of Groverian geodesics for both quantum state vectors and quantum probability amplitudes. In the first case, we show that horizontal affinely parametrized geodesic paths on the Hilbert space of normalized vectors emerge from the minimization of the length specified by the Fubini-Study metric on the manifold of Hilbert space rays. In the second case, we demonstrate that geodesic paths for probability amplitudes arise by minimizing the length expressed in terms of the Fisher information. In both derivations, we find that geodesic equations ar…
We study a class of dynamical systems for which the motions can be described in terms of geodesics on a manifold (ordinary potential models can be cast into this form by means of a conformal map). It is rigorously proven that the geodesic deviation equation of Jacobi, constructed with a second covariant derivative, is unitarily equivalent to that of a parametric harmonic oscillator, and we study the second quatization of this oscillator. The excitations of the Fock space modes correspond to the emission and absorption of quanta into the dynamical medium, thus associating unstable behavior of t…
Bifurcation of closed orbits of Hamiltonian systems with application to geodesics of the Schwarzschild metric
该研究结合Moser不变曲线定理与Poincaré-Birkhoff不动点定理，证明两自由度近可积哈密顿系统中周期非退化不变环面会分岔出无穷多闭轨道，并给出非退化条件的拱线角导数表示，将其应用于中心力问题与Schwarzschild度规赤道测地线扰动研究。
This paper defines a class of variational problems on Lie groups that admit involutive automorphisms. The maximum Principle of optimal control then identifies the appropriate left invariant Hamiltonians on the Lie algebra of the group. The corresponding Hamiltonian equations admit a spectral representation with important applications to the theory of integrable systems. Particular cases make a link to Moser's work relating the solutions of Kepler's problem to the geodesics on spaces of constant curvature, Newmann's mechanical problem on the sphere and related geodesic problem of Jacobi on an e…
Sympletic reduction of the sub-Riemannian geodesic flow for metabelian nilpotent groups
本文研究导子群为阿贝尔群的幂零李群上的次黎曼正规哈密顿流，通过辛约化建立正规轨迹与多项式哈密顿量的对应，并给出可积性判据及全局长度极小化条件。
We study the change of the Laughlin states under large deformations of the geometry of the sphere and the plane, associated with Mabuchi geodesics on the space of metrics with Hamiltonian $S^1$-symmetry. For geodesics associated with the square of the symmetry generator, as the geodesic time goes to infinity, the geometry of the sphere becomes that of a thin cigar collapsing to a line and the Laughlin states become concentrated on a discrete set of $S^1$--orbits, corresponding to Bohr-Sommerfeld orbits of geometric quantization. The lifting of the Mabuchi geodesics to the bundle of quantum sta…
We study geodesics on hypersurfaces close to the standard (n-1)-dimensional sphere in n-dimensional Euclidean space. Following Poincaré, we treat the problem within the framework of the analytical mechanics, and employ the perturbation theory with the view of obtaining a topological classification of the set of geodesics on a manifold. To that end we use the X-ray transform familiar in the integral geometry, and obtain the system of averaged equations of motion, which turns out to be a Hamiltonian one. The system serves an asymptotic reduction of the initial exact system of 2n-2 equations to t…
We study a family of fermionic extensions of the Camassa-Holm equation. Within this family we identify three interesting classes: (a) equations, which are inherently hamiltonian, describing geodesic flow with respect to an H^1 metric on the group of superconformal transformations in two dimensions, (b) equations which are hamiltonian with respect to a different hamiltonian structure and (c) supersymmetric flow equations. Classes (a) and (b) have no intersection, but the intersection of classes (a) and (c) gives a candidate for a new supersymmetric integrable system. We demonstrate the Painlevé…
Motivated by various results on homogeneous geodesics of Riemannian spaces, we study homogeneous trajectories, i.e. trajectories which are orbits of a one-parameter symmetry group, of Lagrangian and Hamiltonian systems. We present criteria under which an orbit of a one-parameter subgroup of a symmetry group G is a solution of the Euler-Lagrange or Hamiltonian equations. In particular, we generalize the `geodesic lemma' known in Riemannian geometry to Lagrangian and Hamiltonian systems. We present results on the existence of homogeneous trajectories of Lagrangian systems. We study Hamiltonian a…
A note on the geodesic deviation equation for null geodesics in the Schwarzschild black-hole
本文利用史瓦西时空测地线方程的哈密顿表述和Morales-Ramis定理，建立了测地线可积性与变分方程微分伽罗瓦可积性之间的联系，并具体展示了零测地线及其变分方程的特例。
In this paper we clarify and generalise previous work by Moser and Belbruno concerning the link between the motions in the classical Kepler problem and geodesic motion on spaces of constant curvature. Both problems can be formulated as Hamiltonian systems and the phase flow in each system is characterised by the value of the corresponding Hamiltonian and one other parameter (the mass parameter in the Kepler problem and the curvature parameter in the geodesic motion problem). Using a canonical transformation the Hamiltonian vector field for the geodesic motion problem is transformed into one wh…
We study the integrability of the geodesic equations of the Chazy- Curzon space-time. It was established that for the equilibrium point $p_ρ=p_z=z=0$ and, $ρ_0 \in (1,\, 2)$, there are only periodic solutions, the Hamiltonian system, describing geodesic motion of Chazy-Curzon space-time has no additional analytic first integral. Our approach is based on the following: if the system has a family of periodic solutions around an equilibrium and if the period function is infinitely branched then the system has no additional analytical first integral.
The link between 3D spaces with (in general, non-constant) curvature and quantum deformations is presented. It is shown how the non-standard deformation of a sl(2) Poisson coalgebra generates a family of integrable Hamiltonians that represent geodesic motions on 3D manifolds with a non-constant curvature that turns out to be a function of the deformation parameter z. A different Hamiltonian defined on the same deformed coalgebra is also shown to generate a maximally superintegrable geodesic motion on 3D Riemannian and (2+1)D relativistic spaces whose sectional curvatures are all constant and e…
We consider magnetic geodesic flows on the 2-torus. We prove that the question of existence of polynomial in momenta first integrals on one energy level leads to a Semi-Hamiltonian system of quasi-linear equations, i.e. in the hyperbolic regions the system has Riemann invariants and can be written in conservation laws form.
The Eisenhart geometric formalism, which transforms an Euclidean natural Hamiltonian $H=T+V$ into a geodesic Hamiltonian ${\cal T}$ with one additional degree of freedom, is applied to the four families of quadratically superintegrable systems with multiple separability in the Euclidean plane. Firstly, the separability and superintegrability of such four geodesic Hamiltonians ${\cal T}_r$ ($r=a,b,c,d$) in a three-dimensional curved space are studied and then these four systems are modified with the addition of a potential ${\cal U}_r$ leading to ${\cal H}_r={\cal T}_r +{\cal U}_r$. Secondly, w…
Analytic Liouville Integrable Geodesic Flow Has Zero Entropy
本文研究实解析刘维尔可积哈密顿系统的拓扑熵，通过有限正则展开与齐格林主导项论证的遍历版本，证明满足条件的哈密顿流及闭实解析黎曼流形上的可积测地流具有零拓扑熵，肯定了帕特纳因1992年的猜想。
We study reductions of the Hamiltonian flows restricted to their invariant submanifolds. As examples, we consider partial Lagrange-Routh reductions of the natural mechanical systems such as geodesic flows on compact Lie groups and $n$-dimensional variants of the classical Hess-Appel'rot case of a heavy rigid body motion about a fixed point.
We explore the dynamical stability of the minisuperspace Hamiltonian of the Bianchi IX cosmological models, giving a gauge-invariant and unapproximated description of the full continuous dynamics, achieved through a geometrical description of the equations of motion in the framework of the theory of Finsler Spaces. The numerical integrations of the geodesics and geodesic deviation equations show clearly the absence of any "traditional" signature of Chaos, while suggesting a chaotic scattering dynamics scenario.
Generalizing symplectic topology from 1 to 2 dimensions
本文将辛流形与 Hamilton 方程从测地线的 1 维框架推广到曲面调和映射的 2 维情形，建立一类类 Hamilton 偏微分方程，并证明非挤压定理及余切丛二次 Hamilton 量的 cuplength 结果。
Geodesic motion on the symplectic leaf of $SO(3)$ with distorted $e(3)$ algebra and Liouville integrability of a free rigid body
该研究计算了$SO(3)$流形对应辛叶上不依赖惯性张量的Poisson结构的显式形式，确认其异于$e(3)$代数，并用所得Poisson括号证明自由刚体的刘维尔可积性，给出Euler-Poisson方程的通解形式。
An infinite family of quasi-maximally superintegrable Hamiltonians with a common set of (2N-3) integrals of the motion is introduced. The integrability properties of all these Hamiltonians are shown to be a consequence of a hidden non-standard quantum sl(2,R) Poisson coalgebra symmetry. As a concrete application, one of this Hamiltonians is shown to generate the geodesic motion on certain manifolds with a non-constant curvature that turns out to be a function of the deformation parameter z. Moreover, another Hamiltonian in this family is shown to generate geodesic motions on Riemannian and rel…
Classical and quantum Hamiltonian reductions of free geodesic systems of complete Riemannian manifolds are investigated. The reduced systems are described under the assumption that the underlying compact symmetry group acts in a polar manner in the sense that there exist regularly embedded, closed, connected submanifolds meeting all orbits orthogonally in the configuration space. Hyperpolar actions on Lie groups and on symmetric spaces lead to families of integrable systems of spin Calogero-Sutherland type.
We show that the standard Heisenberg algebra of quantum mechanics admits a noncommutative differential calculus $Ω^1$ depending on the Hamiltonian $p^2/2m + V(x)$, and a flat quantum connection $\nabla$ with torsion such that a previous quantum-geometric formulation of flow along autoparallel curves (or `geodesics') is exactly Schrödinger's equation. The connection $\nabla$ preserves a generalised `skew metric' given by the canonical symplectic structure lifted to a certain rank (0,2) tensor on the extended phase space where we adjoin a time variable. We also apply the same approach to the Kle…
Translating the dynamics of the Henon--Heiles hamiltonian as a geodesic flow on a Finsler manifold, we obtain a local and synthetic Geometric Indicator of Chaos (GIC) for two degrees of freedom continuous dynamical systems. It represents a link between local quantities and asymptotic behaviour of orbits giving a strikingly evident, one-to-one, correspondence between geometry and instability.
Unified Framework for Geodesic Dynamics with Conservative, Dissipative, and GUP Effects
本文提出统一框架，用于研究包含保守、耗散和GUP修正的测地动力学，通过外部势能诱导速度依赖修正，探讨时空曲率与普朗克尺度物理的相互作用。
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